Structural Theory Notes, Exercises of Theory of Structures

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STRUCTURAL DESIGN I

Felix V. Garde, Jr., msce

Basic Theory:

Shear Stresses in an Uncracked Elastic Beam of Rectangular Cross Section

Derivation of Shear Formula

stresses on subelement.

FBD of subelement.

The normal stresses, σ 1 and σ 2 are,

M y

I

(M + dM )y

I

The normal forces, F 1 & F 2 are,

F 1 =

σ 1 dA =

M y

I

dA

F 2 =

σ 2 dA =

(M + dM )y

I

dA

Derivation of Shear Formula

stresses on subelement.

FBD of subelement.

Apply equation of equilibrium,

ΣF x = 0

F 1 − F 2 + F 3 = 0

substitute the values of F 1 and F 2 ,

and simplify

F 3 =

(dM )y

I

dA

F 3 =

dM

I

ydA

The force F 3 is also equal to

F 3 = τ bdx

Distribution of Shear Stresses in a Rectangular Beam

cross section of beam.

distribution of shear stress.

The first moment Q of the shaded part is

Q =

y dA =

∫ (^) h/ 2

y 1

yb dA

b 2

h^2 4

− y 12

the shear formula , becomes

τ =

V

2 I

h^2 4

− y 12

Shear Stresses in an Uncracked Elastic Beam

Normal, shear, and principal stresses in a homogeneous uncracked beam.

Principal compressive stress trajectories and inclined cracks.

Crack Patterns tn RC beam.

Two types of cracks in rectangular beam with longitudinal flexural reinforcement but no shear reinforcement.

1. Vertical cracks , due to flexural stress.
2. Inclined cracks , due to combined shear and flexure, commonly referred to as inclined cracks, shear cracks, or diagonal tension cracks.

Average Shear Stress between Cracks

Calculation of average shear stress between cracks.

The equilibrium of the shaded por- tion as isolated in (c), is

v =

∆T

b w ∆x or

v =

V

b w jd

The ACI design procedures arbi- trarily replaces jd = d, giving

v =

V

b w d

Beam Action and Arch Action

From,

∆T =

V ∆x jd

it was assumed that the beam is pris- matic and the moment arm, jd is con- stant. In general,

V =

d dx

(T jd)

can be evaluated as

V = T

d dx

(jd) + jd

d dx

(T )

Two extreme cases can be identified.

• Lever arm jd remains constant, then d dx

(jd) = 0 and V = jd

d dx

(T )

• Shear

d(T ) dx

= 0, thus

V = T

d dx

(jd)

or

V = C

d dx

(jd)

Criteria for formation of Diagonal Cracks:

• At a location of large shear force V and small bending moment M , there will be little flexural cracking, if any, prior to the development of a diagonal tension crack. The average shear stress prior to crack formation is,

v =

V

b w d

• If flexural stress are negligibly small at the particular location, the diagonal tensile stresses are inclined at about 45 ◦^ and are numerically equal to the shear stresses, with a maximum at the neutral axis. Consequently, diagonal cracks form mostly at or near the neutral axis and propagate from that location. Web-shear cracks are formed when the diagonal tension stress becomes equal to the tensile strength of the concrete.

Web-shear cracking is relatively rare and occurs mostly near supports of deep, thin-webbed beams or at inflection points of continuous beams.

Behavior of Beams without shear reinforcement

Diagonal tension cracking in reinforced concrete beams.

Inclined cracks and shear reinforcement

Behavior of Beams without shear reinforcement

Loaded beam.

Moment at cracking and failure.